2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59698We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees counted only non-degenerate roots and even then gave much larger bounds, e.g., 248832 via a famous general result of Khovanski. Our bound is sharp, allows real exponents, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive new bounds on the number of real connected components of fewnomial hypersurfaces.18 pages, submitted for publication. Further streamlining and clarifications made. There is also a new figure illustrating the optimality of a variant of Rolles' Theorem we useCombinatoricsAlgebraic GeometryPrimary 34C08; Secondary 14P05,30C15text